Introduction to Bayesian Statistics

Preface xiii 1 Introduction to Statistical Science 1 1.1 The Scientic Method: A Process for Learning 3 1.2 The Role of Statistics in the Scientic Method 5 1.3 Main Approaches to Statistics 5 1.4 Purpose and Organization of This Text 8 2 Scientic Data Gathering 13 2.1 Sampling from a Real Population 14 2.2 Observational Studies and Designed Experiments 17 Monte Carlo Exercises 23 3 Displaying and Summarizing Data 31 3.1 Graphically Displaying a Single Variable 32 3.2 Graphically Comparing Two Samples 39 3.3 Measures of Location 41 3.4 Measures of Spread 44 3.5 Displaying Relationships Between Two or More Variables 46 3.6 Measures of Association for Two or More Variables 49 Exercises 52 4 Logic, Probability, and Uncertainty 59 4.1 Deductive Logic and Plausible Reasoning 60 4.2 Probability 62 4.3 Axioms of Probability 64 4.4 Joint Probability and Independent Events 65 4.5 Conditional Probability 66 4.6 Bayes' Theorem 68 4.7 Assigning Probabilities 74 4.8 Odds and Bayes Factor 75 4.9 Beat the Dealer 76 Exercises 80 5 Discrete Random Variables 83 5.1 Discrete Random Variables 84 5.2 Probability Distribution of a Discrete Random Variable 86 5.3 Binomial Distribution 90 5.4 Hypergeometric Distribution 92 5.5 Poisson Distribution 93 5.6 Joint Random Variables 96 5.7 Conditional Probability for Joint Random Variables 100 Exercises 104 6 Bayesian Inference for Discrete Random Variables 109 6.1 Two Equivalent Ways of Using Bayes' Theorem 114 6.2 Bayes' Theorem for Binomial with Discrete Prior 116 6.3 Important Consequences of Bayes' Theorem 119 6.4 Bayes' Theorem for Poisson with Discrete Prior 120 Exercises 122 Computer Exercises 126 7 Continuous Random Variables 129 7.1 Probability Density Function 131 7.2 Some Continuous Distributions 135 7.3 Joint Continuous Random Variables 143 7.4 Joint Continuous and Discrete Random Variables 144 Exercises 147 8 Bayesian Inference for Binomial Proportion 149 8.1 Using a Uniform Prior 150 8.2 Using a Beta Prior 151 8.3 Choosing Your Prior 154 8.4 Summarizing the Posterior Distribution 158 8.5 Estimating the Proportion 161 8.6 Bayesian Credible Interval 162 Exercises 164 Computer Exercises 167 9 Comparing Bayesian and Frequentist Inferences for Proportion 169 9.1 Frequentist Interpretation of Probability and Parameters 170 9.2 Point Estimation 171 9.3 Comparing Estimators for Proportion 174 9.4 Interval Estimation 175 9.5 Hypothesis Testing 178 9.6 Testing a One-Sided Hypothesis 179 9.7 Testing a Two-Sided Hypothesis 182 Exercises 187 Monte Carlo Exercises 190 10 Bayesian Inference for Poisson 193 10.1 Some Prior Distributions for Poisson 194 10.2 Inference for Poisson Parameter 200 Exercises 207 Computer Exercises 208 11 Bayesian Inference for Normal Mean 211 11.1 Bayes' Theorem for Normal Mean with a Discrete Prior 211 11.2 Bayes' Theorem for Normal Mean with a Continuous Prior 218 11.3 Choosing Your Normal Prior 222 11.4 Bayesian Credible Interval for Normal Mean 224 11.5 Predictive Density for Next Observation 227 Exercises 230 Computer Exercises 232 12 Comparing Bayesian and Frequentist Inferences for Mean 237 12.1 Comparing Frequentist and Bayesian Point Estimators 238 12.2 Comparing Condence and Credible Intervals for Mean 241 12.3 Testing a One-Sided Hypothesis about a Normal Mean 243 12.4 Testing a Two-Sided Hypothesis about a Normal Mean 247 Exercises 251 13 Bayesian Inference for Di erence Between Means 255 13.1 Independent Random Samples from Two Normal Distributions 256 13.2 Case 1: Equal Variances 257 13.3 Case 2: Unequal Variances 262 13.4 Bayesian Inference for Dierence Between Two Proportions Using Normal Approximation 265 13.5 Normal Random Samples from Paired Experiments 266 Exercises 272 14 Bayesian Inference for Simple Linear Regression 283 14.1 Least Squares Regression 284 14.2 Exponential Growth Model 288 14.3 Simple Linear Regression Assumptions 290 14.4 Bayes' Theorem for the Regression Model 292 14.5 Predictive Distribution for Future Observation 298 Exercises 303 Computer Exercises 312 15 Bayesian Inference for Standard Deviation 315 15.1 Bayes' Theorem for Normal Variance with a Continuous Prior 316 15.2 Some Specic Prior Distributions and the Resulting Posteriors 318 15.3 Bayesian Inference for Normal Standard Deviation 326 Exercises 332 Computer Exercises 335 16 Robust Bayesian Methods 337 16.1 Eect of Misspecied Prior 338 16.2 Bayes' Theorem with Mixture Priors 340 Exercises 349 Computer Exercises 351 17 Bayesian Inference for Normal with Unknown Mean and Variance 355 17.1 The Joint Likelihood Function 358 17.2 Finding the Posterior when Independent Jeffreys' Priors for mu and sigma2 Are Used 359 17.3 Finding the Posterior when a Joint Conjugate Prior for mu and sigma2 Is Used 361 17.4 Difference Between Normal Means with Equal Unknown Variance 367 17.5 Difference Between Normal Means with Unequal Unknown Variances 377 Computer Exercises 383 Appendix: Proof that the Exact Marginal Posterior Distribution of mu is Student's t 385 18 Bayesian Inference for Multivariate Normal Mean Vector 393 18.1 Bivariate Normal Density 394 18.2 Multivariate Normal Distribution 397 18.3 The Posterior Distribution of the Multivariate Normal Mean Vector when Covariance Matrix Is Known 398 18.4 Credible Region for Multivariate Normal Mean Vector when Covariance Matrix Is Known 400 18.5 Multivariate Normal Distribution with Unknown Covariance Matrix 402 Computer Exercises 406 19 Bayesian Inference for the Multiple Linear Regression Model 411 19.1 Least Squares Regression for Multiple Linear Regression Model 412 19.2 Assumptions of Normal Multiple Linear Regression Model 414 19.3 Bayes' Theorem for Normal Multiple Linear Regression Model 415 19.4 Inference in the Multivariate Normal Linear Regression Model 419 19.5 The Predictive Distribution for a Future Observation 425 Computer Exercises 428 20 Computational Bayesian Statistics Including Markov Chain Monte Carlo 431 20.1 Direct Methods for Sampling from the Posterior 436 20.2 Sampling - Importance - Resampling 450 20.3 Markov Chain Monte Carlo Methods 454 20.4 Slice Sampling 470 20.5 Inference from a Posterior Random Sample 473 20.6 Where to Next? 475 A Introduction to Calculus 477 B Use of Statistical Tables 497 C Using the Included Minitab Macros 523 D Using the Included R Functions 543 E Answers to Selected Exercises 565 References 591 Index 595